Abstract

By taking a counterexample, we prove that the
multistep iteration process is faster than the Mann and Ishikawa iteration
processes for Zamfirescu operators.

1. Introduction

Let be a nonempty convex subset of a normed space , and let be a mapping.

(a) For arbitrary , the sequence defined by
where is a sequence in , is known as the Mann iteration process [1].

(b) For arbitrary , the sequence defined by
where and are sequences in , is known as the Ishikawa iteration process [2].

(c) For arbitrary , the sequence defined by
where and , , are sequences in and denoted by , is known as the multistep iteration process [3].

Definition 1 (see [4]). Suppose that and are two real convergent sequences with limits and , respectively. Then, is said to converge faster than if

Definition 2 (see [4]). Let and be two fixed-point iteration procedures which, both, converge to the same fixed point , say, with error estimates,
where . If converges faster than , then is said to converge faster than .

Theorem 3 (see [5]). Let be a complete metric space, and let be a mapping for which there exist real numbers , , and satisfying and such that, for each pair , at least one of the following is true:,
,
. Then, has a unique fixed point , and the Picard iteration defined by
converges to for any .

Remark 4. An operator , which satisfies the contraction conditions ()–() of Theorem 3, will be called a Zamfirescu operator [4, 6, 7].In [6, 7], Berinde introduced a new class of operators on a normed space satisfying
for any , , and .He proved that this class is wider than the class of Zamfirescu operators.The following results are proved in [6, 7].

Theorem 5 (see [7]). Let be a nonempty closed convex subset of a normed space . Let be an operator satisfying (4). Let be defined through the iterative process and . If and , then converges strongly to the unique fixed point of .

Theorem 6 (see [6]). Let be a nonempty closed convex subset of an arbitrary Banach space , and let be an operator satisfying (4). Let be defined through the iterative process and , where and are sequences of positive numbers in with satisfying . Then, converges strongly to the fixed point of .

Theorem 7. Let be a closed convex subset of an arbitrary Banach space . Let the Mann and Ishikawa iteration processes with real sequences and satisfy , , and . Then, and converge strongly to the unique fixed point of . Let be a Zamfirescu operator, and, moreover, the Mann iteration process converges faster than the Ishikawa iteration process to the fixed point of .

Theorem 8. Let be a closed convex subset of an arbitrary Banach space , and let be a Zamfirescu operator. Let be defined by and with a sequence in satisfying . Then, converges strongly to the fixed point of , and, moreover, the Picard iteration converges faster than the Mann iteration.

Remark 9. In [9], Qing and Rhoades by taking a counterexample showed that the Mann iteration process converges more slowly than the Ishikawa iteration process for Zamfirescu operators.

In this paper, we establish a general theorem to approximate fixed points of quasi-contractive operators in a Banach space through the multistep iteration process. Our result generalizes and improves upon, among others, the corresponding results of Babu and Vara Prasad [8] and Berinde [4, 6, 7].

We also prove that the Mann iteration process and the Ishikawa iteration process converge more slowly than the multistep iteration process for Zamfirescu operators.

2. Main Results

We now prove our main results.

Theorem 10. Let be a nonempty closed convex subset of an arbitrary Banach space , and let be an operator satisfying (4). Let be defined through the iterative process and , where and , (), are sequences in with . If , then is a singleton, and the sequence converges strongly to the fixed point of .

Proof. Assume that and . Then, using , we have
Now, for and , (4) gives
By substituting (6) in (5), we obtain
In a similar fashion, again by using , we can get
where and
It can be easily seen that, for , we have
Substituting (9) in (10) gives us
It may be noted that, for and , the following inequality is always true:
From (11) and (12), we get
By repeating the same procedure, finally from (7) and (10), we yield
By (14), we inductively obtain
Using the fact that , , and , it results that
which, by (15), implies that
Consequently, , and this completes the proof.

Now, by a counterexample, we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

Example 11. Suppose that is defined by ; , , and ; , , and . It is clear that is a Zamfirescu operator with a unique fixed point and that all of the conditions of Theorem 10 are satisfied. Also, , . Suppose that . For the Mann and Ishikawa iteration processes, we have
where implies that
Now, consider
It is easy to see that
Hence,
Thus, the Mann iteration process converges more slowly than the multistep iteration process to the fixed point of .Similarly,
with
implies that
Thus, the Ishikawa iteration process converges more slowly than the multistep iteration process to the fixed point of .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the referees for their useful comments and suggestions. This study was supported by research funds from Dong-A University.